Problem: Zachary paid for a $\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used?
Answer: If $p$ is the number of pennies and $n$ is the number of nickels that Zachary used, then we are given \begin{align*}
p+n&=32\text{, and} \\
p+5n&=100.
\end{align*} Subtracting the first equation from the second, we find $4n=68$ which implies $n=\boxed{17}$.